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Algebra / Geometry II: Unit 6- Right Triangles SUCCESS CRITERIA: 1. Find the third side of a right triangle using Pythagorean’s Theorem. 2. Determine if a triangle is acute, right, or obtuse using Pythagorean's Theorem. 3. Determine the sides of special right triangles. 4. Find a side or angle of a right triangle using trigonometry. 5. Use trigonometry to answer real world problems. INSTRUCTOR: Craig Sherman PMI-NJ Center for Teaching & Learning Hidden Lake High School Westminster Public Schools ~1 NJCTL.org EMPOWER TARGET MA.10.G.08.04 SCALE THEME Trigonometric Ratios PROFICIENCY SCALE: SCORE 4.0 3.5 3.0 2.0 1.0 REQUIREMENTS In addition to exhibiting Score 3.0 performance, in-depth inferences and applications that go BEYOND what was taught in class. Score 4.0 does not equate to more work but rather a higher level of performance. In addition to Score 3.0 performance, in-depth inferences and applications with partial success. The learner exhibits no major errors or omissions regarding any of the information and processes (simple or complex) that were explicitly taught. o Find the third side of a right triangle using Pythagorean’s Theorem., AND o Determine if a triangle is acute, right, or obtuse using Pythagorean's Theorem , AND o Determine the sides of special right triangles, AND o Find a side or angle of a right triangle using trigonometry, AND o Use trigonometry to answer real world problems. Can do one or more of the following skills / concepts: There are no major errors or omissions regarding the simpler details and processes as the learner… o Find the third side of a right triangle using Pythagorean’s Theorem, OR o Determine if a triangle is acute, right, or obtuse using Pythagorean's Theorem, OR o Determine the sides of special right triangles, OR o Find a side or angle of a right triangle using trigonometry, OR o Use trigonometry to answer real world problems Know and use the vocabulary Identify the Basic Elements With help, a partial understanding of some of the simpler details and process PMI-NJ Center for Teaching & Learning ~2 NJCTL.org Similarity in Right Triangles WORD or CONCEPT DEFINITION or NOTES EXAMPLE or GRAPHIC REPRESENTATION geometric mean INSTRUCTION 1: SOPHIA: GEOMETRIC MEAN INSTRUCTION 2: GEOMETRIC MEAN OF RIGHT TRIANGLE EXEMPLARS: GEOMETRIC MEAN with NUMBERS: GEOMETRIC MEAN with RIGHT TRIANGLES: GIVEN: GIVEN: 12 and 16 STEP 1: Proportion STEP 2: geometric mean STEP 3: numbers` STEP 4: cross multiply STEP 5: solve STEP 6: square root = 𝑥 12 = 𝑥 𝑥 = 16 (x)(x) = (12)(16) x2 = 192 x ≈ 13.8 𝑥 STEP 1: Proportion STEP 2: geometric mean = 8 = 8 (altitude) STEP 3: Parts Hypotenuse STEP 4: cross multiply STEP 5: solve STEP 6: divide PMI-NJ Center for Teaching & Learning ~3 16 8 = 𝑥 (8)(8) = (16)(x) 64 = 16x x=4 8 NJCTL.org Classwork 1. Find the geometric mean of the pairs. Write the answer is simplest radical form. a. 8 and 32 c. 10 and 15 b. 2 and 24 d. 14 and 18 2. GE is the altitude of triangle DEF. What are the three similar triangles? Solve for x. 3. 4. 5. 6. 7. 8. PMI-NJ Center for Teaching & Learning ~4 NJCTL.org Homework 9. Find the geometric mean of the pairs. Write the answer is simplest radical form. a. 9 and 25 b. 6 and 54 c. 12 and 20 d. 10 and 28 10. SQ is the altitude of triangle PQR. What are the three similar triangles? Solve for x. 11. 12. 13. 14. 15. PMI-NJ Center for Teaching & Learning NJCTL.org ~1 Pythagorean Theorem and its Converse WORD or CONCEPT DEFINITION or NOTES EXAMPLE or GRAPHIC REPRESENTATION leg hypotenuse Pythagorean Theorem INSTRUCTION 2: SOPHIA TUTORIAL INSTRUCTION 1: KHAN ACADEMY Leg2 + Leg2 = Hypotenuse2 a 2 + b2 = c 2 ( Integrated II – Rt. Triangle Trig ~1~ )2 + ( )2 = ( )2 NJCTL.org Class work 16. In the triangle above, the hypotenuse is side ______, the legs are sides _____ and ______. Refer the diagram below to answer questions 17-22. Write the answer in simplest radical form. 17. If AB = 9 and BC = 12, then AC = _________. 18. If AB = 11 and AC = 61, then BC = _________. 19. If BC = 45 and AC = 53, then AB = __________. 20. If AC = 9 and AB = 3, then BC = _________. 21. If BC = 14 and AC = 8√7, then AB = __________. 22. If AB = 2√3 and BC = 6√5, then AC = _________. Classify the sides as lengths of an acute, right, obtuse, or not a triangle. 23. 8, 11, 5 24. 20, 29, 21 25. 6, 2, 2√2 26. 9, 11√2, 7√5 27. Find the perimeter and area of the rectangle. Integrated II – Rt. Triangle Trig ~1~ NJCTL.org Homework 28. In the triangle above the hypotenuse is side ______, the legs are sides _____ and ______. Refer the diagram below to answer questions 29-34. Write the answer in simplest radical form. 29. If AB = 33 and BC = 56, then AC = _________. 30. If AB = 12 and AC = 37, then BC = _________. 31. If BC = 80 and AC = 89, then AB = __________. 32. If AC = 14 and AB = 5√2, then BC = _________. 33. If BC = 9 and AC = 16, then AB = __________. 34. If AB = 5√2 and BC = 7√6, then AC = _________. Classify the sides as lengths of an acute, right, obtuse, or not a triangle. 35. 19, 7, 15 36. 9, 4, 3 37. 14, 14, 14√2 38. 3√11, 4√7, 5√6 39. Find the perimeter and area of the triangle. 31cm 29cm Integrated II – Rt. Triangle Trig ~1~ NJCTL.org Special Right Triangles INSTRUCTION 1: KHAN ACADEMY (30-60-90) INSTRUCTION 2: SOPHIA TUTORIAL INSTRUCTION 3: KHAN ACADEMY (45-45-90) Classwork 40. The length of each leg of an isosceles right triangle is 13 cm. What is the length of the hypotenuse? 41. The length of the hypotenuse of a 45°-45°-90° triangle is 16 cm. What is the length of each leg? 42. The length of the shorter leg of a 30°-60°-90° triangle is 5 cm. What is the length of the longer leg? What is the length of the hypotenuse? 43. The length of the longer leg of a 30°-60°-90° triangle is 12 cm. What is the length of the hypotenuse? What is the length of the shorter leg? 44. The length of the hypotenuse of a 30°-60°-90° triangle is 12 cm. What is the length of the longer leg? What is the length of the shorter leg? Find the lengths of the missing sides. 45. 46. 18 cm y 60° x 60° x 47. x 18 cm y 48. y 60° x 45° y 6 cm 18 cm 49. x 45° 6 cm y 50. Find the area of the triangle. Round to the nearest hundredth. 51. A skateboarder constructs a ramp using plywood. The height of the ramp is 4 feet. If the ramp falls at a 60° angle, what is the length of the plywood Integrated II – Rt. Triangle Trig ~1~ NJCTL.org Homework 52. The length of each leg of an isosceles right triangle is 9 cm. What is the length of the hypotenuse? 53. The length of the hypotenuse of a 45°-45°-90° triangle is 10 cm. What is the length of each leg? 54. The length of the shorter leg of a 30°-60°-90° triangle is 13 cm. What is the length of the longer leg? What is the length of the hypotenuse? 55. The length of the longer leg of a 30°-60°-90° triangle is 21 cm. What is the length of the hypotenuse? What is the length of the shorter leg? 56. The length of the hypotenuse of a 30°-60°-90° triangle is 21 cm. What is the length of the longer leg? What is the length of the shorter leg? Find the lengths of the missing sides. 57. 24 cm y 60° 58. y x x 59. 24 cm 60° 60. x y 60° 10 cm y 45° x 24 cm 61. x 45° 10 cm y 62. Find the area of the triangle. Round to the nearest hundredth. 63. A skateboarder constructs a ramp using plywood. The length of the plywood is 7 feet long and falls at a 40° angle. What is the height of the ramp? Integrated II – Rt. Triangle Trig ~1~ NJCTL.org Trigonometric Ratios WORD or CONCEPT DEFINITION or NOTES EXAMPLE or GRAPHIC REPRESENTATION trigonometry reference angle side opposite side adjacent sine cosine tangent Integrated II – Rt. Triangle Trig ~1~ NJCTL.org INSTRUCTION 1: KHAN ACADEMY INTRODUCTION INSTRUCTION 2: KHAN BASIC TRIG Integrated II – Rt. Triangle Trig ~1~ NJCTL.org Class work Refer to the triangle below to answer questions 65-66. 64. side opposite to ∠W is _______side adjacent to ∠W is _______ side opposite to∠B is _______side adjacent to ∠B is _______hypotenuse is _______. 64. sinW = ______ sinB = ______ cosW = ______ tanW = _______. cosB = ______ tanB = _______. Evaluate. Round the nearest ten-thousandth. 65. sin90o 67. tan25o 66. cos52o 68. tan88o 69. If the sin30 = .5, the cos60= ______________. Why? 70. If the cos25= .9063, the sin65=_____________. Why? Homework Refer to the triangle below to answer questions 75-76. 71. side opposite to ∠W is _______side adjacent to ∠W is _______ side opposite to ∠B is _______side adjacent to ∠B is _______hypotenuse is _______. 72. sinW = ______ sinB = ______ cosW = ______ tanW = _______. cosB = ______ tanB = _______. Evaluate. Round the nearest ten-thousandth. 73. sin45o 75. cos69o 74. tan 77o 76. tan33o 77. If the sin60 = .8660, the cos30= ______________. Why? 78. If the cos65= .4226, the sin25=_____________. Why? Integrated II – Rt. Triangle Trig ~1~ NJCTL.org Solving Right Triangles NOTES: 2-sides __________ __________ = 3-function 1-ref./ _____ O A HA INSTRUCTION 1: SOPHIA TUTORIAL Class work Solve the triangle. Round to the nearest hundredth. 79. BC = ______ 80. STEP 1: Reference ∠ ________ 270 = STEP 2: Sides ________ 270 = 𝑥 STEP 3: Function tan 270 = STEP 4: Solve 13 * tan 270 = STEP 5: Solution 𝑥 𝑂 13 𝐴 13 𝑥 6.62 m∠D = ______ 81. m∠I = ______ 82. JL = ______ Integrated II – Rt. Triangle Trig ~1~ NJCTL.org Homework Solve the triangle. Round to the nearest hundredth. 83. BC = ______ 84. m∠D = ______ 63. m∠I = ______ 85. JL = ______ Integrated II – Rt. Triangle Trig ~1~ NJCTL.org Angle of Elevation and Depression VOCABULARY WORD or CONCEPT DEFINITION or NOTES EXAMPLE or GRAPHIC REPRESENTATION angle of elevation angle of depression NOTES: Angle of Elevation = Angle of Depression INSTRUCTION 1: SOPHIA TUTORIAL Integrated II – Rt. Triangle Trig ~1~ NJCTL.org Class work 86. Angela flies a kite at a 60o angle of elevation. The kite’s string is 275 feet. Angela’s arm is 4.5 feet off the ground. How high is the kite off the ground? (Round to the nearest hundredth) 87. An airplane flies 6 miles above the ground. The distance from the start of the runway to the airplane is 15 miles. What is the angle of depression the airplane must use to land? (Round to the nearest hundredth) 88. You are looking at the top of a tree. The angle of elevation is 78o. The distance from the top of the tree to your position is 125 feet. If you are 5.25 feet tall, how far are you from the base of the tree? (Round to the nearest hundredth) Homework 89. An airplane is 20 miles from the start of the runway. The angle of depression the airplane must use to land is 40 o. How high is the airplane above the ground? (Round to the nearest hundredth) 90. You are standing on a mountain that is 6842 feet high. You look down at your campsite at angle of 38 o. If you are 5.6 feet tall, how far is the base of the mountain from the campsite? (Round to the nearest hundredth) 91. You are looking at the top of a tree. The angle of elevation is 66o. The distance from the top of the tree to your position is 108 feet. If you are 5.75 feet tall, how tall is the tree? (Round to the nearest hundredth) Integrated II – Rt. Triangle Trig ~1~ NJCTL.org Right Triangles & Trigonometry Unit Review Multiple Choice - Choose the correct answer for each question. No partial credit will be given. 1. Find the length of the missing side of the triangle. Write the answer in simplest radical form. a. 117 2. 15√2 2 c. 18.76 d. √352 b. right c. obtuse d. not a triangle c. 10√3 b. 7.5 d. 9√13 Find the value of x. Write the answer in simplest radical form. a. 4√2 6. b. 16√22 Find the value of x. Write the answer in simplest radical form. a. 5. d. 3√13 Tell whether the lengths7√2, 3√7, and 5 form the sides of an acute, right, obtuse, or not a triangle. a. acute 4. c. √117 Find the length of the missing side of the triangle. Write the answer in simplest radical form. a. 4√22 3. b. 9√13 b. 8√2 c. 8√2 2 d. 4 Find the value of x. Write the answer in simplest radical form. a. 2√2 Integrated II – Rt. Triangle Trig b. 4√2 ~1~ c. 4√2 2 d. 2 NJCTL.org 7. In the triangle below, what is the side opposite to ∠J. a. JR 8. d. LR b. 1 c. 2 5 7 d. 7 5 Find the m∠V. Round the answer to the nearest hundredth. a. 33.69o 10. c. JL What is tan K? a. 2 9. b. LJ b. 56.31o c. 41.81o d. 48.19o ̅̅̅̅. Round the answer to the nearest hundredth. Find the length of DT a. 8.31 b. 6.64 Integrated II – Rt. Triangle Trig c. 6.26 ~1~ d. 3.99 NJCTL.org 11. .In the figure, sin 47 = 15/x, which of the following equations is also true? x 15 47° a. sin 43 = x/15 b. cos 43 = 15/x c. cos 47 = 15/x d. tan 47 = x/15 12. Find the value of x. Write the answer in simplest radical form. 15 30° x a. 15√2 c. 10√3 b. 7.5 2 d. 9√13 13. Find the m∠D. Round the answer to the nearest hundredth. a. 50.71o b. 39.29o 14. Find the geometric mean of 5 and 9. a. 3√5 b. 7 c. 35.10o d. 54.90o c. 5√3 d. 9√3 c. 5√7 d. 25 15. Solve for x. 16 x a. 7 9 b. 12 Integrated II – Rt. Triangle Trig ~1~ NJCTL.org 16. In the figure, sin 47 = 15/x, which of the following equations is also true? 15 x 47° a. sin 43 = x/15 b. cos 43 = 15/x c. cos 47 = 15/x d. tan 47 = x/15 Short Constructed Response - Write the answer for each question. No partial credit will be given. 17. Tell whether the lengths 7√3, 3√11, and 4√5 form the sides of an acute, right, obtuse or not a triangle. 18. Michael is flying a kite at an angle of 45o. The kite's string is 202 feet long and Amy arm is 4 feet off the ground. How high is the kite off the ground? 19. Find the perimeter & area of the triangle. Round your answers to the nearest hundredth. Integrated II – Rt. Triangle Trig ~1~ NJCTL.org 20. If cos18° = .9511, then what is the sin72°? Explain the relationship between these two values and why the relationship occurs. Extended Constructed Response – Solve the problem, showing all work. Partial credit may be given. 21. You are on the deck of a cruise ship that is 425 m above the water. You look down towards the ocean and see dolphins. The angle of depression from your line of sight to the dolphins is 42° & 44°, respectively. a) How far is the first dolphin from the ship? b) How far is the second dolphin from the ship? c) What is the distance between the 2 dolphins? d) If the cruise ship stops at one of its destinations & the dolphins continue to travel toward the ship at a rate of 11 meters per second, how long will it take them to get to the ship? Integrated II – Rt. Triangle Trig ~1~ NJCTL.org